Nonlinear Problems in Elasticity

Course description

Chiara Daraio The course is an advanced introduction to nonlinear problems of elastic continuum media. It will start with a review of basic principles and equations for linear elasticity in continuum media under the small strains assumptions. It will then introduce slender structures, i.e. rods and plates and analyze buckling and instabilities introduced by the geometric nonlinearity of their kinematics. Bringing kinematic nonlinearities to three dimensional to continuum structures will naturally lead to finite strains and nonlinear elasticity. The theoretical topics will be illustrated by work-out examples and experiments using rods, strings, rubber balloons and sheets. The concepts are introduced using precise mathematical deductions and physical assumptions and will be discussed in depth. However, calculations will be kept to a minimum by using symbolic and numeric computations to solve different problem settings. The presented method can be applied for various engineering applications, e.g. the prediction of the failure of structures by buckling and instability, the design of novel materials with architectured microstructures or the analysis of soft biological materials.

Time and place

Tuesday, 10:15 - 13:00 (Lecture&Exercise),LEE C 104

Lecturer

Prof. Dr. Andrei Constantinescu
Open-door policy or by appointment, CLA J 27

Teaching assistant

Jinwoong Cha
Open-door policy or by appointment, CLA J 15.1

 

Syllabus

Date Lecture Exercise
23.02. Lecture #1: Motivation, Euler Elastica Calculus of variations
Slide
Note
Mathematica notebook
Exercise #1: Calculus of variations
Exercise #1
Further reading
01.03. Lecture #2:Linear Elasticity: basic equations and exact solutions
Slide
Note
Mathematica notebook
Exercise #2: Mechanics of curly hair
Exercise #2
08.03. Lecture #3:Structires in Equilibrium
Note
15.03. Lecture #4:Reciprocity or further variaonal calculus & Energy in Linear Elascity
Slide
Exercise #4
22.03. Lecture #5:Finite strains, Nonlinear elasticity, Inflating a rubber ballon
Slide
05.04. Lecture #6:Review_Stress_LargeStrain, Prestrain_Torsion
Note_large strain
Note_torsion
Paper
12.04. Lecture #7:Rubber ballons and Stability
Slide
Note in French
Paper
19.04. Lecture #8:Rubber ballons and Stability 2
Slide
26.04. Lecture #9:Physical mechanism for material elasticity
03.05. Lecture #10:Nonlinear elasticity: basic equations and exact solutions
10.05. Lecture #11:Stretch of a rubber sheets
17.05. Lecture #12:Torsional instability
24.05. Lecture #13:Stability of many rubber balloons
31.05. Lecture #14: From instabilities to phase change: shape memory alloys

 

Literature:

  1. Basile Audoly and Yves Pomeau Elasticity and geometry: from hair curls to the non-linear response of shells, Oxford University Press, 2010
  2. Patrick Ballard and Alain Millard Poutres et arcs elastiques, Editions Ecole Polytechnique, 2009
  3. Andrei Constantinescu and Alexander Korsunsky Elasticity with Mathematica : An Introduction to Continuum Mechanics and Linear Elasticity , Cambridge University Press, 2007
  4. Yibin B Fu and Raymond W OgdenNonlinear elasticity: theory and applications, Cambridge University Press, 2001
  5. Gerhard A HolzapfelNonlinear solid mechanics, Wiley, 2000
  6. I Mueller and P StrehlowRubber and Rubber Balloons (Lecture Notes in Physics 637),Springer: Heidelberg,Germany, 2004

 

Performance assessment

Homework: 30%
Project work: 30%
Final Exam: 40%

 

Student project: